5.3 solving systems of linear equations by elimination · section 5.3 solving systems of linear...

216 Chapter 5 Systems of Linear Equations

5.3

How can you use elimination to solve a

system of linear equations?

Work with a partner. Solve each system of linear equations by using two methods.

Method 1: Subtract.

Subtract Equation 2 from Equation 1. What is the result? Explain how you can use the result to solve the system of equations.

Method 2: Add.

Add the two equations. What is the result? Explain how you can use the result to solve the system of equations.

Is the solution the same using both methods?

a. 2x + y = 4 b. 3x − y = 4 c. x + 2y = 7

2x − y = 0 3x + y = 2 x − 2y = − 5

ACTIVITY: Using Elimination to Solve a System11

Work with a partner.

2x + y = 2 Equation 1

x + 5y = 1 Equation 2

a. Can you add or subtract the equations to solve the system of linear equations? Explain.

b. Explain what property you can apply to Equation 1 in the system so that the y-coeffi cients are the same.

c. Explain what property you can apply to Equation 2 in the system so that the x-coeffi cients are the same.

d. You solve the system in part (b). Your partner solves the system in part (c). Compare your solutions.

e. Use a graphing calculator to check your solution.

ACTIVITY: Using Elimination to Solve a System22

Solving Systems of Linear Equations by Elimination

Systems of EquationsIn this lesson, you will● write and solve systems

of linear equations by elimination.

● solve real-life problems.

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Section 5.3 Solving Systems of Linear Equations by Elimination 217

Work with a partner. Solve the puzzle to fi nd the name of a famous mathematician who lived in Egypt around 350 A.D.

ACTIVITY: Solving a Secret Code33

Use what you learned about systems of linear equations to complete Exercises 4 – 6 on page 221.

4. IN YOUR OWN WORDS How can you use elimination to solve a system of linear equations?

5. STRUCTURE When can you add or subtract equations in a system to solve the system? When do you have to multiply fi rst? Justify your answers with examples.

6. LOGIC In Activity 2, why can you multiply equations in the system by a constant and not change the solution of the system? Explain your reasoning.

x 2y 52x y 5

2x y 3x y 0

3x 3y 02x 2y 8

x y 4x y 2x y 2

2x 2y 4

2x y 0x y 3

x y 5x y 1

B W R M F Y K N

O J A S I D X Z

Q P C E G B T J

M R C Z N O U W

K X U H L Y S Q

F E A S W K R M

G J Z N H V D G

E

4

3

2

1

0

1

2

3 L X L F Q O B

3 2 1 0 1 2 3 4

Find Entry PointsWhat is the fi rst thing you do to solve a system of linear equations? Why?

Math Practice



Lesson5.3Lesson Tutorials

Check

Equation 1

x + 3y = − 2

7 + 3(− 3) =? − 2

− 2 = − 2 ✓Equation 2

x − 3y = 16

7 − 3(− 3) =? 16

16 = 16 ✓

EXAMPLE Solving a System of Linear Equations by Elimination11

Solve the system by elimination. x + 3y = −2 Equation 1

x − 3y = 16 Equation 2

Step 1: The coeffi cients of the y-terms are already opposites.

Step 2: Add the equations.

x + 3y = − 2 Equation 1

x − 3y = 16 Equation 2

2x = 14 Add the equations.

Step 3: Solve for x.

2x = 14 Equation from Step 2

x = 7 Divide each side by 2.

Step 4: Substitute 7 for x in one of the original equations and solve for y.

x + 3y = − 2 Equation 1

7 + 3y = − 2 Substitute 7 for x.

3y = − 9 Subtract 7 from each side.

y = − 3 Divide each side by 3.

The solution is (7, − 3).

Solve the system of linear equations by elimination. Check your solution.

1. 2x − y = 9 2. − 5x + 2y = 13 3. 3x + 4y = − 6

4x + y = 21 5x + y = − 1 7x + 4y = − 14

Exercises 7–12

Solving a System of Linear Equations by Elimination

Step 1: Multiply, if necessary, one or both equations by a constant so at least 1 pair of like terms has the same or opposite coeffi cients.

Step 2: Add or subtract the equations to eliminate one of the variables.

Step 3: Solve the resulting equation for the remaining variable.

Step 4: Substitute the value from Step 3 into one of the original equations and solve.

Study TipBecause the coeffi cients of x are the same, youcan also solve the system by subtracting in Step 2.

x + 3y = − 2 x − 3y = 16 6y = − 18So, y = − 3.



EXAMPLE Solving a System of Linear Equations by Elimination22

Solve the system by elimination. −6x + 5y = 25 Equation 1

−2x − 4y = 14 Equation 2

Step 1: Multiply Equation 2 by 3.

− 6x + 5y = 25 − 6x + 5y = 25 Equation 1

− 2x − 4y = 14 − 6x − 12y = 42 Revised Equation 2

Step 2: Subtract the equations.

− 6x + 5y = 25 Equation 1

− 6x − 12y = 42 Revised Equation 2

17y = − 17 Subtract the equations.

Step 3: Solve for y.

17y = − 17 Equation from Step 2

y = − 1 Divide each side by 17.

Step 4: Substitute − 1 for y in one of the original equations and solve for x.

− 2x − 4y = 14 Equation 2

− 2x − 4(− 1) = 14 Substitute − 1 for y.

− 2x + 4 = 14 Multiply.

− 2x = 10 Subtract 4 from each side.

x = − 5 Divide each side by − 2.

The solution is (− 5, − 1).


4. 3x + y = 11 5. 4x − 5y = − 19 6. 5y = 15 − 5x

6x + 3y = 24 − x − 2y = 8 y = − 2x + 3

Study TipIn Example 2, notice that you can also multiply Equation 2 by − 3 and then add the equations.

Multiply by 3.

Check

1010

10

10

10

2x 4y 14

6x 5y 25



EXAMPLE Real-Life Application33You buy 8 hostas and 15 daylilies for $193. Your friend buys 3 hostas and 12 daylilies for $117. Write and solve a system of linear equations to fi nd the cost of each daylily.

Use a verbal model to write a system of linear equations.

Numberof hostas

⋅ Cost of each

hosta, x + Number of

daylilies ⋅

Cost of each daylily, y

= Total cost

The system is: 8x + 15y = 193 Equation 1 (You)

3x + 12y = 117 Equation 2 (Your friend)

Step 1: To fi nd the cost y of each daylily, eliminate the x-terms. Multiply Equation 1 by 3. Multiply Equation 2 by 8.

8x + 15y = 193 24x + 45y = 579 Revised Equation 1

3x + 12y = 117 24x + 96y = 936 Revised Equation 2

Step 2: Subtract the revised equations.

24x + 45y = 579 Revised Equation 1

24x + 96y = 936 Revised Equation 2

− 51y = − 357 Subtract the equations.

Step 3: Solving the equation − 51y = − 357 gives y = 7.

So, each daylily costs $7.

7. A landscaper buys 4 peonies and 9 geraniums for $190. Another landscaper buys 5 peonies and 6 geraniums for $185. Write and solve a system of linear equations to fi nd the cost of each peony.

Exercises 16–21

Multiply by 3.

Methods for Solving Systems of Linear Equations

Method When to Use

Graphing (Lesson 5.1) To estimate solutions

Substitution (Lesson 5.2)

When one of the variables in one of the equations has a coeffi cient of 1 or − 1

Elimination (Lesson 5.3)

When at least 1 pair of like terms has the same or opposite coeffi cients

Elimination (Multiply First) (Lesson 5.3)

When one of the variables cannot be eliminated by adding or subtracting the equations

Multiply by 8.



Exercises5.3

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

1. WRITING Describe how to solve a system of linear equations by elimination.

2. NUMBER SENSE When should you use multiplication to solve a system of linear equations by elimination?

3. WHICH ONE DOESN’T BELONG? Which system of equations does not belong with the other three? Explain your reasoning.

3x + 3y = 3

2x − 3y = 7

− 2x + y = 6

2x − 3y = − 10

2x + 3y = 11

3x − 2y = 10

x + y = 5

3x − y = 3

Use a method from Activity 1 to solve the system.

4. x + y = 3 5. − x + 3y = 0 6. 3x + 2y = 3

x − y = 1 x + 3y = 12 3x − 2y = − 9


7. x + 3y = 5 8. x − 2y = − 7 9. 4x + 3y = − 5

− x − y = − 3 3x + 2y = 3 − x + 3y = − 10

10. 2x + 7y = 1 11. 2x + 5y = 16 12. 3x − 2y = 4

2x − 4y = 12 3x − 5y = − 1 6x − 2y = − 2

13. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations.

14. RAFFLE TICKETS You and your friend are selling raffl e tickets for a new laptop. You sell 14 more tickets than your friend sells. Together, you and your friend sell 58 tickets.

a. Write a system of linear equations that represents this situation.

b. How many tickets does each of you sell?

15. JOGGING You can jog around your block twice and the park once in 10 minutes. You can jog around your block twice and the park 3 times in 22 minutes.


b. How long does it take you to jog around the park?

Help with Homework

11

5x + 2y = 9 Equation 1 3x − 2y = −1 Equation 2 2x = 10 x = 5 The solution is (5, −8).

✗

y Elimination 22




16. 2x − y = 0 17. x + 4y = 1 18. − 2x + 3y = 7

3x − 2y = − 3 3x + 5y = 10 5x + 8y = − 2

19. 3x + 3 = 3y 20. 2x − 6 = 4y 21. 5x = 4y + 8

2x − 6y = 2 7y = − 3x + 9 3y = 3x − 3

22. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations.

23. REASONING For what values of a and b should you solve the system by elimination?

a. 4x − y = 3 b. x − 7y = 6

ax + 10y = 6 − 6x + by = 9

Determine whether the line through the fi rst pair of points intersects the line through the second pair of points. Explain.

24. Line 1: (− 2, 1), (2, 7) 25. Line 1: (3, − 2), (7, − 1)

Line 2: (− 4, − 1), (0, 5) Line 2: (5, 2), (6, − 2)

26. AIRPLANES Two airplanes are fl ying to the same airport. Their positions are shown in the graph. Write a system of linear equations that represents this situation. Solve the system by elimination to justify your answer.

27. TEST PRACTICE The table shows the number of correct answers on a practice standardized test. You score 86 points on the test, and your friend scores 76 points.


b. How many points is each type of question worth?

You Your Friend

Multiple Choice 23 28

Short Response 10 5

x + y = 1 Equation 1 −5x + 5y = −5 5x + 3y = −3 Equation 2 5x + 3y = −3 8y = −8 y = −1

The solution is (2, −1).

✗ Multiply by −5.

22 33

x

y

3

5

6

7

8

4

9

10

11

12

2

1

04 5 6 7 8 9 10 11 12 13 14 153210

Airport

j y y

3

4

2

1

03210



Decide whether the two equations are equivalent. (Section 1.2 and Section 1.3)

34. 4n + 1 = n − 8 35. 2a + 6 = 12 36. 7v − 3

— 2

= 5

3n = − 9 a + 3 = 6 14v − 3 = 15

37. MULTIPLE CHOICE Which line has the same slope as y = 1

— 2

x − 3? (Section 4.4)

○A y = − 2x + 4 ○B y = 2x + 3 ○C y − 2x = 5 ○D 2y − x = 7

28. LOGIC You solve a system of equations in which x represents the number of adult tickets sold and y represents the number of student tickets sold. Can (− 6, 24) be the solution of the system? Explain your reasoning.

29. VACATION The table shows the activities of two tourists at a vacation resort. You want to go parasailing for 1 hour and horseback riding for 2 hours. How much do you expect to pay?

30. REASONING The solution of a system of linear equations is (2, − 4). One equation in the system is 2x + y = 0. Explain how you could fi nd a second equation for the system. Then fi nd a second equation. Solve the system by elimination to justify your answer.

31. JEWELER A metal alloy is a mixture of two or more metals. A jeweler wants to make 8 grams of 18-carat gold, which is 75% gold. The jeweler has an alloy that is 90% gold and an alloy that is 50% gold. How much of each alloy should the jeweler use?

32. PROBLEM SOLVING A powerboat takes 30 minutes to travel 10 miles downstream. The return trip takes 50 minutes. What is the speed of the current?

33. Solve the system of equations by elimination.

2x − y + 3z = − 1

x + 2y − 4z = − 1

y − 2z = 0

Parasailing Horseback Riding Total Cost

Tourist 1 2 hours 5 hours $205

Tourist 2 3 hours 3 hours $240

O4)yoa sy

mix

28.

29.

30. REASO(2, − 4−how yfi nd ajustify

31 JEWELER A metal alloy is a ma

V I D E O

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5.3 solving systems of linear equations by elimination · section 5.3 solving systems of linear...

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